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Theorem resspsradd 17882
Description: A restricted power series algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
Hypotheses
Ref Expression
resspsr.s  |-  S  =  ( I mPwSer  R )
resspsr.h  |-  H  =  ( Rs  T )
resspsr.u  |-  U  =  ( I mPwSer  H )
resspsr.b  |-  B  =  ( Base `  U
)
resspsr.p  |-  P  =  ( Ss  B )
resspsr.2  |-  ( ph  ->  T  e.  (SubRing `  R
) )
Assertion
Ref Expression
resspsradd  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  U ) Y )  =  ( X ( +g  `  P ) Y ) )

Proof of Theorem resspsradd
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 resspsr.u . . 3  |-  U  =  ( I mPwSer  H )
2 resspsr.b . . 3  |-  B  =  ( Base `  U
)
3 eqid 2467 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
4 eqid 2467 . . 3  |-  ( +g  `  U )  =  ( +g  `  U )
5 simprl 755 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  X  e.  B )
6 simprr 756 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  B )
71, 2, 3, 4, 5, 6psradd 17846 . 2  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  U ) Y )  =  ( X  oF ( +g  `  H
) Y ) )
8 resspsr.s . . . 4  |-  S  =  ( I mPwSer  R )
9 eqid 2467 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
10 eqid 2467 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
11 eqid 2467 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
12 fvex 5876 . . . . . . . 8  |-  ( Base `  R )  e.  _V
13 resspsr.2 . . . . . . . . . 10  |-  ( ph  ->  T  e.  (SubRing `  R
) )
14 resspsr.h . . . . . . . . . . 11  |-  H  =  ( Rs  T )
1514subrgbas 17250 . . . . . . . . . 10  |-  ( T  e.  (SubRing `  R
)  ->  T  =  ( Base `  H )
)
1613, 15syl 16 . . . . . . . . 9  |-  ( ph  ->  T  =  ( Base `  H ) )
17 eqid 2467 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
1817subrgss 17242 . . . . . . . . . 10  |-  ( T  e.  (SubRing `  R
)  ->  T  C_  ( Base `  R ) )
1913, 18syl 16 . . . . . . . . 9  |-  ( ph  ->  T  C_  ( Base `  R ) )
2016, 19eqsstr3d 3539 . . . . . . . 8  |-  ( ph  ->  ( Base `  H
)  C_  ( Base `  R ) )
21 mapss 7462 . . . . . . . 8  |-  ( ( ( Base `  R
)  e.  _V  /\  ( Base `  H )  C_  ( Base `  R
) )  ->  (
( Base `  H )  ^m  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin } )  C_  (
( Base `  R )  ^m  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin } ) )
2212, 20, 21sylancr 663 . . . . . . 7  |-  ( ph  ->  ( ( Base `  H
)  ^m  { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin } ) 
C_  ( ( Base `  R )  ^m  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
2322adantr 465 . . . . . 6  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( ( Base `  H
)  ^m  { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin } ) 
C_  ( ( Base `  R )  ^m  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
24 eqid 2467 . . . . . . 7  |-  ( Base `  H )  =  (
Base `  H )
25 eqid 2467 . . . . . . 7  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
26 reldmpsr 17821 . . . . . . . . . 10  |-  Rel  dom mPwSer
2726, 1, 2elbasov 14541 . . . . . . . . 9  |-  ( X  e.  B  ->  (
I  e.  _V  /\  H  e.  _V )
)
2827ad2antrl 727 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( I  e.  _V  /\  H  e.  _V )
)
2928simpld 459 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  I  e.  _V )
301, 24, 25, 2, 29psrbas 17841 . . . . . 6  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  B  =  ( ( Base `  H )  ^m  { f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
318, 17, 25, 9, 29psrbas 17841 . . . . . 6  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( Base `  S )  =  ( ( Base `  R )  ^m  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
3223, 30, 313sstr4d 3547 . . . . 5  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  B  C_  ( Base `  S
) )
3332, 5sseldd 3505 . . . 4  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  X  e.  ( Base `  S ) )
3432, 6sseldd 3505 . . . 4  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  ( Base `  S ) )
358, 9, 10, 11, 33, 34psradd 17846 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  S ) Y )  =  ( X  oF ( +g  `  R
) Y ) )
3614, 10ressplusg 14600 . . . . . . 7  |-  ( T  e.  (SubRing `  R
)  ->  ( +g  `  R )  =  ( +g  `  H ) )
3713, 36syl 16 . . . . . 6  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  H ) )
3837adantr 465 . . . . 5  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( +g  `  R )  =  ( +g  `  H
) )
39 ofeq 6527 . . . . 5  |-  ( ( +g  `  R )  =  ( +g  `  H
)  ->  oF
( +g  `  R )  =  oF ( +g  `  H ) )
4038, 39syl 16 . . . 4  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  oF ( +g  `  R )  =  oF ( +g  `  H
) )
4140oveqd 6302 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X  oF ( +g  `  R
) Y )  =  ( X  oF ( +g  `  H
) Y ) )
4235, 41eqtrd 2508 . 2  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  S ) Y )  =  ( X  oF ( +g  `  H
) Y ) )
43 fvex 5876 . . . . 5  |-  ( Base `  U )  e.  _V
442, 43eqeltri 2551 . . . 4  |-  B  e. 
_V
45 resspsr.p . . . . 5  |-  P  =  ( Ss  B )
4645, 11ressplusg 14600 . . . 4  |-  ( B  e.  _V  ->  ( +g  `  S )  =  ( +g  `  P
) )
4744, 46mp1i 12 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( +g  `  S )  =  ( +g  `  P
) )
4847oveqd 6302 . 2  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  S ) Y )  =  ( X ( +g  `  P ) Y ) )
497, 42, 483eqtr2d 2514 1  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  U ) Y )  =  ( X ( +g  `  P ) Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113    C_ wss 3476   `'ccnv 4998   "cima 5002   ` cfv 5588  (class class class)co 6285    oFcof 6523    ^m cmap 7421   Fincfn 7517   NNcn 10537   NN0cn0 10796   Basecbs 14493   ↾s cress 14494   +g cplusg 14558  SubRingcsubrg 17237   mPwSer cmps 17811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6903  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-fsupp 7831  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-sca 14574  df-vsca 14575  df-tset 14577  df-subg 16012  df-rng 17014  df-subrg 17239  df-psr 17816
This theorem is referenced by:  subrgpsr  17885  ressmpladd  17930
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